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Balaban 11-cage

3-regular graph

Field	Value
name	Balaban 11-cage
image	[[Image:Balaban 11-cage.svg	240px]]
image_caption	The Balaban 11-cage
namesake	Alexandru T. Balaban
vertices	112
edges	168
automorphisms	64
girth	11
radius	6
diameter	8
chromatic_number	3
chromatic_index	3
independence_number	52
properties	Cubic
Cage
Hamiltonian

Cage Hamiltonian In the mathematical field of graph theory, the Balaban 11-cage or Balaban (3,11)-cage is a 3-regular graph with 112 vertices and 168 edges named after Alexandru T. Balaban.

The Balaban 11-cage is the unique (3,11)-cage. It was discovered by Balaban in 1973. The uniqueness was proved by Brendan McKay and Wendy Myrvold in 2003.

The Balaban 11-cage is a Hamiltonian graph and can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.

It has independence number 52, chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

The characteristic polynomial of the Balaban 11-cage is: :(x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2\cdot :\cdot(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 \cdot :\cdot(x^5+x^4-8 x^3-6 x^2+12 x+4)^8.

The automorphism group of the Balaban 11-cage is of order 64.

Gallery

Image:balaban_11-cage_3COL.svg|The chromatic number of the Balaban 11-cage is 3. Image:balaban_11-cage_3color_edge.svg|The chromatic index of the Balaban 11-cage is 3. Image: balaban_11-cage_alternative_drawing.svg|Alternative drawing of the Balaban 11-cage.

References

{{citation

References

"Balaban 11-Cage".
[[Alexandru Balaban. Balaban, Alexandru T.]], ''Trivalent graphs of girth nine and eleven, and relationships among cages'', Revue Roumaine de Mathématiques Pures et Appliquées '''18''' (1973), 1033-1043. {{MR. 0327574
"Cage Graph".
Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008)
{{harvtxt. Heal. 2016
[[Peter Eades. P. Eades]], J. Marks, [[Petra Mutzel. P. Mutzel]], S. North. "Graph-Drawing Contest Report", TR98-16, December 1998, Mitsubishi Electric Research Laboratories.

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